so he's King because they rebeled against the former King and overthrew him?

Yes. It will come out why they rebelled, but that's the gist of it. Daenarys and her brother are descendants of the king they deposed. That's why Robert was talking about having her killed in the last episode.

__________________In this world of sin and sorrow there is always something to be thankful for; as for me, I rejoice that I am not a Republican.
- H. L. Mencken